It is not possible to find a frame of reference where a photon is at rest. I will argument in two different ways:
1. Maxwell equations and electromagnetic argument:
From Maxwell it is expected that electromagnetic disturbances propagate in vacuum at a constant speed c~299792458 m/s which is the maximum speed for the propagation of electromagnetic interactions.
If you could find a rest frame for a photon (i.e. a frame of reverence where the speed of photons is zero), then, in this frame of reference any electromagnetic interaction would be impossible (as photons are the carriers of the electromagnetic interaction). For example, the force between two electrons at rest would be $F=0$ for any location of the electrons as the field would not be able to propagate between them. This is absurd, and therefore it is not possible to find a frame of reference where a photon is at rest.
2. Corpuscular nature of photons and Quantum Mechanics:
The energy $E$ of a photon is defined as $E=hf$ where $h$ is Plank's constant and $f$ stands for the photon's frequency but $c = \lambda f$ (with $\lambda$ being the wavelength). This product can be zero in three different ways:
- $\lambda = 0$, $f$ finite. In this case, the photon has zero wavelength and therefore infinite momentum and finite energy which is absurd.
- $f = 0$, $\lambda$ finite. In this case, the photon has no energy but a finite momentum ($p = h/\lambda$) which is again absurd.
- $\lambda = 0$ and $f = 0$. The photon has zero frequency (zero energy) and zero wavelength (infinite momentum) which is double absurd.
Therefore both Classical Electromagnetism and Quantum Theory of Light deny the possibility of a frame of reference where a photon can be found at rest.
